Cos Calculator

Free Online Cos Calculator — Solve cos(θ) Instantly

This cos calculator (cosine calculator) is a complete trigonometry tool that solves cosine problems in every direction you're likely to face — plug in an angle to get cos(θ), plug in the adjacent side and hypotenuse of a right triangle to get the angle, or plug in a decimal cosine value to solve an equation like cos(θ) = 0.5 for θ. Every calculation is backed by a to-scale diagram and a full step-by-step solution, so you can see exactly how the cosine value or angle was derived instead of just getting a bare number.

Whether you're a student solving trigonometry homework, checking a physics or engineering problem involving angles and forces, or just need a quick sine, cosine, and tangent value for an angle in degrees or radians, this calculator covers the full sin, cos, tan relationship in one place.

What Is Cosine (cos) in Trigonometry?

Cosine is one of the three primary trigonometric ratios, alongside sine and tangent. For any angle θ measured in a right triangle, cosine is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse — the longest side, opposite the right angle. This ratio, written cos(θ) = Adjacent ÷ Hypotenuse, is the foundation of the SOH-CAH-TOA mnemonic taught in every trigonometry class:

• SOH — Sine = Opposite ÷ Hypotenuse
• CAH — Cosine = Adjacent ÷ Hypotenuse
• TOA — Tangent = Opposite ÷ Adjacent

The Cosine Formula and the Unit Circle

Beyond right triangles, cosine is extended to any angle — even angles greater than 90° or negative angles — using the unit circle, a circle of radius 1 centered at the origin. As the angle θ sweeps counter-clockwise from the positive x-axis, the point where the angle's ray meets the circle has coordinates (cos θ, sin θ). In other words, cos(θ) is simply the x-coordinate of that point. This is exactly what the diagram above shows: the orange dashed line marks the x-distance from the y-axis out to the point on the circle, and that distance is the cosine value.

• cos(θ) = Adjacent ÷ Hypotenuse (right-triangle definition)
• cos(θ) = x-coordinate of the point on the unit circle at angle θ
• θ = cos⁻¹(x) — the inverse cosine (arccos) function, used to solve for the angle when the ratio is known
• cos(θ) is always between -1 and 1, since it's a ratio on a circle of radius 1

How to Use This Cos Calculator — Step by Step

Pick the mode that matches your problem, enter your numbers, and read the result along with the diagram and worked solution below it.

• cos(θ) from an angle: type an angle and choose degrees, radians, or gradians — the calculator instantly returns cos(θ), sin(θ), tan(θ), the quadrant, and the reference angle, plotted on a unit circle.
• θ from right-triangle sides: enter the adjacent side and hypotenuse of a right triangle — the calculator applies cos(θ) = Adjacent ÷ Hypotenuse, then uses the inverse cosine to solve for θ, and draws the triangle to scale with every side labeled.
• θ from a cosine value: enter any decimal between -1 and 1 to solve equations like cos(θ) = 0.5 for θ, returning the principal angle plus the second solution between 0° and 360°.

Worked Example — Solving cos(60°)

Suppose you need cos(60°). Switch to the first mode, enter 60, choose degrees, and the calculator converts 60° to π/3 radians, locates the point on the unit circle, and reads off the x-coordinate: cos(60°) = 0.5 exactly. The diagram shows the radius line reaching a point exactly halfway across the circle's width, matching the well-known result cos(60°) = 1/2.

Now suppose you're given a right triangle with an adjacent side of 4 and a hypotenuse of 5 — a classic 3-4-5 triangle. Switching to the second mode and entering those values gives cos(θ) = 4 ÷ 5 = 0.8, and applying the inverse cosine, θ = cos⁻¹(0.8) ≈ 36.87°. The calculator also finds the missing opposite side using the Pythagorean theorem: √(5² − 4²) = 3, completing the full 3-4-5 right triangle.

Common Cosine Values Table

These standard angles come up constantly in trigonometry, and it helps to recognize them at a glance rather than recalculating them each time:

• cos(0°) = 1
• cos(30°) = √3/2
• cos(45°) = √2/2
• cos(60°) = 1/2
• cos(90°) = 0
• cos(180°) = -1
• cos(270°) = 0
• cos(360°) = 1

Understanding the Sign of cos(θ) by Quadrant

Because the unit circle wraps all the way around, cos(θ) changes sign depending on which quadrant the angle lands in. This calculator automatically detects and displays the quadrant for any angle you enter:

• Quadrant 1 (0°-90°): cos(θ) is positive — the point is to the right of the y-axis, above the x-axis
• Quadrant 2 (90°-180°): cos(θ) is negative — the point moves to the left of the y-axis
• Quadrant 3 (180°-270°): cos(θ) is still negative — the point stays left of the y-axis, below the x-axis
• Quadrant 4 (270°-360°): cos(θ) is positive again — the point returns to the right of the y-axis

Cosine, Sine, and Tangent — How They Relate

Cosine never works in isolation — it's closely tied to sine and tangent, and this calculator shows all three whenever you solve for an angle. Sine is the y-coordinate on the same unit circle diagram (the green dashed segment), and tangent is simply the ratio tan(θ) = sin(θ) ÷ cos(θ). A useful identity worth remembering is sin²(θ) + cos²(θ) = 1, which follows directly from the Pythagorean theorem applied to the unit circle's radius of 1.

Degrees vs Radians vs Gradians

Angles can be measured in more than one unit, and this calculator supports the three most common ones. Degrees split a full circle into 360 equal parts and are the most familiar unit in everyday geometry. Radians measure angle by arc length relative to the radius, making a full circle equal to 2π radians — this is the unit calculus and most programming languages use internally. Gradians (or gons) split a full circle into 400 parts and appear mostly in surveying and some European engineering contexts. Whichever unit your problem uses, simply select it from the dropdown and the calculator handles the conversion automatically.

Where Cosine Calculations Are Used in Real Life

Cosine isn't just a classroom exercise — it shows up anywhere angles and directions interact:

• Physics: resolving forces along an axis, calculating work done (Work = Force × Distance × cos θ), and analyzing wave interference.
• Engineering and construction: calculating horizontal load components, ramp run lengths, and structural bracing angles.
• Navigation and GPS: the Law of Cosines uses cosine to calculate distances between coordinates on a sphere.
• Computer graphics and game development: rotating and projecting 3D objects onto a 2D screen relies heavily on cosine.
• Signal processing: cosine waves are the building block of the Fourier transform, used in audio compression and image processing (like JPEG).

Tips for Getting Accurate Results

Always double-check which angle unit your problem is stated in before entering a value — a common mistake is entering a radian value while the unit dropdown is still set to degrees, which will give a wildly different (and wrong) answer. When solving for an angle from a cosine value or triangle sides, remember that inverse cosine only returns one principal solution between 0° and 180°; if your problem expects an angle between 0° and 360°, check the second solution shown in the results, since cosine repeats and multiple angles can share the same cosine value.